Threshold solutions for the focusing $L^{2}$ -supercritical NLS Equations

TL;DR

This paper analyzes the behavior of solutions to the focusing $L^2$-supercritical nonlinear Schrödinger equation at a critical threshold, extending previous results and classifying solutions using modulation theory.

Contribution

It provides a detailed classification of solutions at the critical threshold for the focusing $L^2$-supercritical NLS, generalizing earlier findings to broader cases.

Findings

Classified solutions at the critical threshold using modulation theory

Extended previous results to $L^2$-supercritical regimes

Identified solution behaviors at the threshold level

Abstract

We investigate the $L^2$-supercritical and $\dot{H}^1$-subcritical nonlinear Schr\"{o}dinger equation in $H^1$. In \cite{G1} and \cite{yuan}, the mass-energy quantity $M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present paper, we study the dynamics at the critical level $M(u)^{\frac{1-s_{c}}{s_{c}}}E(u)=M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ and classify the corresponding solutions using modulation theory, non-trivially generalize the results obtained in \cite{holmer3} for the 3D cubic Schr\"{o}dinger equation.